Find $a$ such that $ax^2+12x+9$ is the square of a binomial.
Answer: The square of the binomial $rx+s$ is  \[(rx+s)^2=r^2x^2+2rsx+s^2.\]If this is equal to $ax^2+12x+9$, then $s$ must be either 3 or $-3$.  Since $(rx+s)^2=(-rx-s)^2$, we may choose either $s=3$ or $s=-3$, and the solution will be the same.  We choose $s=3$.

The square of $rx+3$ is  \[(rx+3)^2=r^2x^2+6rx+9.\]If this is equal to $ax^2+12x+9$ then we must have $12=6r$ or $r=2$.  This gives our square: \[\left(2x+3\right)^2=4x^2+12x+9.\]Therefore $a=\boxed{4}$.